Statistics on averages and variation Basic Computation Vari

Statistics on averages and variation -

Basic Computation: Variance, Standard Deviation Given the sample data

x: 23 17 15 30 25

a) Find the range.

c) Use the results of part (b) and appropriate computation formulas to computer the sample variance s² and sample standard deviation s.

d) Use the defining formulas to compute the sample variance s² and sample standard deviation s.

Please show work, thank you.

Solution

a)

Range = max - min = 39 - 15 = 15 [answer]

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b)

Consider the table:

c)

variance = [Sum(x^2) - Sum(x)^2/n]/(n-1) = (2568 - 110^2/5)/(5-1) = 37 [ANSWER]

standard deviation = sqrt(37) = 6.08276253 [ANSWER]

d)

Getting the mean, X,          
          
X = Sum(x) / n          
Summing the items, Sum(x) =    110      
As n =    5      
Thus,          
X =    22      
          
Setting up tables,          
x   x - X   (x - X)^2  
23   1   1  
17   -5   25  
15   -7   49  
30   8   64  
25   3   9  
          
          
          
Thus, Sum(x - X)^2 =    148      
          
Thus, as           
          
s^2 = Sum(x - X)^2 / (n - 1)          
          
As n =    5      
          
s^2 =    37   [ANSWER, VARIANCE]  
          
Thus,          
          
s =    6.08276253   [ANSWER, STANDARD DEVIATION]

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e)

sigma^2 = Sum(x - X)^2 / (n - 1) = 148/5 = 29.6 [ANSWER]

sigma = 5.440588203 [ANSWER]